Density Matrix Renormalization Group Calculations Research Articles

Tensor Network State Algorithms on AI Accelerators.

We introduce novel algorithmic solutions for hybrid CPU-multiGPU tensor network state algorithms utilizing non-Abelian symmetries building on AI-motivated state-of-the-art hardware and software technologies. The presented numerical simulations on the FeMo cofactor, which plays a crucial role in converting atmospheric nitrogen to ammonia, are far beyond the scope of traditional approaches. Our large-scale SU(2) spin adapted density matrix renormalization group calculations up to bond dimension D = 216 on complete active space (CAS) size of 18 electrons in 18 orbitals [CAS(18, 18)] demonstrate that the current limit of exact solution, i.e. full-CI limit, can be achieved in fraction of time. Furthermore, benchmarks up to CAS(113, 76) demonstrate the utilization of NVIDIA's highly specialized AI accelerators via NVIDIA Tensor Cores, leading to performance around 115 TFLOPS on a single node supplied with eight NVIDIA A100 devices. As a consequence of reaching 71% of the full capacity of the hardware, the cubic scaling of computational time with bond dimension can be reduced to a linear form for a broad range of D values; thus, breaking the current computational limits of small CAS spaces in ab initio quantum chemistry and material science is becoming a reality. In comparison to strict U(1) implementations with matching accuracy, our solution has an estimated effective performance of 300-500 TFLOPS, which emphasizes the mutual need for both algorithmic and technological developments to push current frontiers on classical computation.

Strong Pairing Originated from an Emergent Z_{2} Berry Phase in La_{3}Ni_{2}O_{7}.

The recent discovery of high-temperature superconductivity in La_{3}Ni_{2}O_{7} offers a fresh platform for exploring unconventional pairing mechanisms. Starting with the basic argument that the electrons in d_{z^{2}} orbitals nearly form local moments, we examine the effect of the Hubbard interaction U on the binding strength of Cooper pairs based on a single-orbital bilayer model with intralayer hopping t_{∥} and interlayer superexchange J_{⊥}. By extensive density matrix renormalization group calculations, we observe a remarkable enhancement in binding energy as much as 10-20 times larger with U/t_{∥} increasing from 0 to 12 at J_{⊥}/t_{∥}∼1. We demonstrate that such a substantial enhancement stems from a kinetic-energy-driven mechanism. Specifically, a Z_{2} Berry phase will emerge at large U due to the Hilbert space restriction (Mottness), which strongly suppresses the mobility of single particle propagation as compared to U=0. However, the kinetic energy of the electrons (holes) can be greatly restored by forming an interlayer spin-singlet pairing, which naturally results in a superconducting state even for relatively small J_{⊥}. An effective hard-core bosonic model is further proposed to estimate the superconducting transition temperature at the mean-field level.

Antiferromagnetic Bosonic t-J Models and Their Quantum Simulation in Tweezer Arrays.

The combination of optical tweezer arrays with strong interactions-via dipole exchange of molecules and Van der Waals interactions of Rydberg atoms-has opened the door for the exploration of a wide variety of quantum spin models. A next significant step will be the combination of such settings with mobile dopants. This will enable one to simulate the physics believed to underlie many strongly correlated quantum materials. Here, we propose an experimental scheme to realize bosonic t-J models via encoding the local Hilbert space in a set of three internal atomic or molecular states. By engineering antiferromagnetic (AFM) couplings between spins, competition between charge motion and magnetic order similar to that in high-T_{c} cuprates can be realized. Since the ground states of the 2D bosonic AFM t-J model we propose to realize have not been studied extensively before, we start by analyzing the case of two dopants-the simplest instance in which their bosonic statistics plays a role-and compare our results to the fermionic case. We perform large-scale density matrix renormalization group calculations on six-legged cylinders, and find a strong tendency for bosonic holes to form stripes. This demonstrates that bosonic, AFM t-J models may contain similar physics as the collective phases in strongly correlated electrons.

Extracting correlation length in Mott insulators by strong-field driving

The breakdown of a Mott-insulator when subjected to intense laser fields is characterized by the formation of doublon-hole pairs. This breakdown is furthermore evidenced by the production of high harmonics that can be experimentally measured. Here, we present an approach for extracting the doublon-hole correlation length of a Mott insulator. The method is based on a dynamical calculation of the Mott insulator’s rate of charge production in response to an applied strong-field laser pulse. We find that coupling the Mott insulator to a metal drastically increases the correlation length, in support of our recent hypothesis [Phys. Rev. B 108, 144434 (2023)2469-995010.1103/PhysRevB.108.144434] that coupling to a metal enhances the charge fluctuations in the insulator. We confirm our conclusions using density matrix renormalization group (DMRG) calculations. The proposed method can be applied to experimentally measured observables, such as differential reflectivity or the high harmonic generation (HHG) spectrum to extract doublon-hole correlation length.

CFT and lattice correlators near an RG domain wall between minimal models

Conformal interfaces separating two conformal field theories (CFTs) provide maps between different CFTs, and naturally exist in nature as domain walls between different phases. One particularly interesting construction of a conformal interface is the renormalization group (RG) domain wall between CFTs. For a given Virasoro minimal model \mathcal{M}_{k+3,k+2}ℳk+3,k+2, an RG domain wall can be generated by a specific deformation which triggers an RG flow towards its adjacent Virasoro minimal model \mathcal{M}_{k+2,k+1}ℳk+2,k+1 with the deformation turned on over part of the space. An algebraic construction of this domain wall was proposed by Gaiotto in [J. High Energy Phys. 12, 103 (2012)]. In this paper, we will provide a study of this RG domain wall for the minimal case k=2k=2, which can be thought of as a nonperturbative check of the construction. In this case the wall is separating the Tricritical Ising Model (TIM) CFT and the Ising Model (IM) CFT. We will check the analytical results of correlation functions from the RG brane construction with the numerical density matrix renormalization group (DMRG) calculation using a lattice model proposed in [arXiv.1206.1332, Science 344, 280 (2014)], and find a perfect agreement. We comment on possible experimental realizations of this RG domain wall.

Emergent Superconductivity and Competing Charge Orders in Hole-Doped Square-Lattice t-J Model.

The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.

On the magnetization of the 120° order of the spin-1/2 triangular lattice Heisenberg model: a DMRG revisited.

We revisit the issue about the magnetization of the 120° order in the spin-1/2 triangular lattice Heisenberg model with density matrix renormalization group (DMRG). The accurate determination of the magnetization of this model is challenging for numerical methods and its value exhibits substantial disparities across various methods. We perform a large-scale DMRG calculation of this model by employing bond dimension as large asD=24000and by studying the system with width as large asLy=12. With careful extrapolation with truncation error and suitable finite size scaling, we give a conservative estimation of the magnetization asM0=0.208(8). The ground state energy per site we obtain isEg=-0.5503(8). Our results provide valuable benchmark values for the development of new methods in the future.

The energy spectrum and magnetization profile of the decorated spin ladder systems

The peculiarities of the energy spectrum of the decorated mixed spin two-leg ladder system having three spin unit cells have been studied by the perturbation theory in the limit of dominant spin coupling in ladder rungs. It is shown, that despite the singlet ground state, this spin ladder may have an intermediate plateau in a low-temperature magnetization profile. The existence of a similar plateau was shown also in the case of some intermediate values of coupling parameters by means of exact diagonalization study of small ladder clusters and by the density-matrix renormalization group calculations. We also found that the Ising type of interactions in ladder rungs may lead to the disappearance of the above intermediate magnetization plateau for the corresponding isotropic spin ladder.

Singlet quantum phases of the frustrated spin-1/2 ladder with ferromagnetic (F) exchange in legs and alternating F-AF exchange in rungs

Ladders with 2N spins-1/2 and singlet (S = 0) ground states are studied numerically using exact diagonalization and density matrix renormalization group (DMRG) calculations. The curvatures E ″(s) of states with the lowest energy E(s) at spin s = S/N ≤ 1 are obtained in the thermodynamic limit. All states have E ″(s) > 0 in two phases, one with strong F exchange J F in rungs, the other with strong F exchange J L in legs. The Dimer phase with frustrated F exchanges has one or more states with E ″(s) < 0. All three phases have finite gaps to the lowest triplet state. The ladder at J F → ∞ is adiabatically connected to the spin-1 Heisenberg AF chain, and at J L → ∞ to the J 1 − J 2 model.

Ground-state phase diagram and universality of sequential topological valence-bond-solid quantum transitions in a mixed tetramer chain

The ground-state ordering of a quantum mixed-spin Heisenberg tetramer chain composed of an alternate sequence of s = 1 and S=3/2 dimers is studied in detail as a function of two considered exchange interactions ascribed to similar and dissimilar spin pairs. At zero magnetic field, the ferrimagnetic mixed spin-(1, 1, 3/2, 3/2) Heisenberg tetramer chain displays, depending on a mutual interplay between two considered exchange interactions, three distinct gapped valence-bond-solid phases separated by gap-closing quantum critical points. Using density-matrix renormalization group calculations we construct the full ground-state phase diagram as a function of the interaction ratio and magnetic field, which exhibits besides three gapped valence-bond-solid phases special Kosterlitz–Thouless and topological quantum critical points. A tangential finite-size scaling analysis is employed to obtain precise estimates of the zero-field valence-bond-solid transitions and unveil their common logarithmic correction to a power-law scaling of the correlation length.

Distributed Multi-GPU Ab Initio Density Matrix Renormalization Group Algorithm with Applications to the P-Cluster of Nitrogenase.

The presence of many degenerate d/f orbitals makes polynuclear transition-metal compounds, such as iron-sulfur clusters in nitrogenase, challenging for state-of-the-art quantum chemistry methods. To address this challenge, we present the first distributed multi-graphics processing unit (GPU) ab initio density matrix renormalization group (DMRG) algorithm suitable for modern high-performance computing (HPC) infrastructures. The central idea is to parallelize the most computationally intensive part─the multiplication of O(K2) operators with a trial wave function, where K is the number of spatial orbitals, by combining operator parallelism for distributing the workload with a batched algorithm for performing contractions on GPU. With this new implementation, we are able to reach an unprecedentedly large bond dimension D = 14,000 on 48 GPUs (NVIDIA A100 80 GB SXM) for an active space model (114 electrons in 73 active orbitals) of the P-cluster, which is nearly 3 times larger than the bond dimensions reported in previous DMRG calculations for the same system using only central processing units (CPUs).

Heteroatom-vacancy centres in molecular nanodiamonds: a computational study of organic molecules possessing triplet ground states through σ-overlap.

Small molecules possessing a triplet ground state are fundamentally intriguing but also in high demand for applications such as quantum sensing and quantum computing. Such molecules are rare, and most examples involve extended π-systems. Topology and shape of the spin density will be very different for molecules where the triplet state arises from σ-overlap. Drawing inspiration from NV- (anionic nitrogen-vacancy) centres in a diamond crystal, which possess triplet ground states that are robust due to the distortion-preventing crystal lattice, we investigate hetero-atom substituted diamondoids (molecular nanodiamonds) as molecular mimics for NV- centres. It is found that even in these small systems, distortions that stabilize singlet states are energetically costly, and the triplet states are more stable than the singlets. The stabilization of the triplet over the singlet is 13, 16, and 18 kcal mol-1, in anionic C3v-C33H36N- and in the charge-neutral molecules C3v-C33H36O and C3v-C33H36S, respectively, using CAM-B3LYP-D3(BJ)/Def2-QZVPP. Comparable numbers are obtained with other density functional theory (DFT) methods, including double-hybrids. Wavefunction-based approaches on the other hand disagree in their predictions: While the MP2 method applied with the DLPNO approximation predicts a preference for the singlet, density matrix renormalization group (DMRG) calculations qualitatively agree with DFT in their prediction of a triplet ground state, although by a small margin, for C3v-C33H36N- and C3v-C33H36O, but not for C3v-C33H36S. Weighing the evidence, we conclude, with reasonable confidence for C3v-C33H36N- and C3v-C33H36O and lesser confidence for C3v-C33H36S, that the ground state for the molecular nanodiamonds studied is a triplet state.

Predicting the FCI Energy of Large Systems to Chemical Accuracy from Restricted Active Space Density Matrix Renormalization Group Calculations.

We theoretically derive and validate with large scale simulations a remarkably accurate power law scaling of errors for the restricted active space density matrix renormalization group (DMRG-RAS) method [J. Phys. Chem. A 126, 9709] in electronic structure calculations. This yields a new extrapolation method, DMRG-RAS-X, which reaches chemical accuracy for strongly correlated systems such as the chromium dimer, dicarbon up to a large cc-pVQZ basis and even a large chemical complex such as the FeMoco with significantly lower computational demands than those of previous methods. The method is free of empirical parameters, performed robustly and reliably in all examples we tested, and has the potential to become a vital alternative method for electronic structure calculations in quantum chemistry and more generally for the computation of strong correlations in nuclear and condensed matter physics.

Trimer quantum spin liquid in a honeycomb array of Rydberg atoms

Quantum spin liquids are elusive but paradigmatic examples of strongly correlated quantum states that are characterized by long-range quantum entanglement. Recently, signatures of a gapped topological Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} spin liquid have been observed in a system of Rydberg atoms; however, the full capability of these platforms to realize quantum spin liquids extends far beyond this state alone. Here, we propose the realization of a different class of spin liquids in a honeycomb array of Rydberg atoms. Exploring the system’s quantum phase diagram using density-matrix renormalization group and exact diagonalization calculations, we identify several density-wave-ordered phases and a trimer spin liquid ground state with an emergent U(1) × U(1) local symmetry. This liquid state originates from superpositions of classical trimer configurations on the dual triangular lattice in the regime where third-nearest-neighbor atoms lie within the Rydberg blockade radius. Finally, we discuss the conditions to enhance the preparation fidelity of this state by a general Rydberg-blockade-based projection mechanism, test the robustness of the trimer spin liquid phase in a range of realistic parameters, and outline methods for its experimental detection.

Auger spectroscopy beyond the ultra-short core-hole relaxation time approximation

We present a time-dependent computational approach to study Auger electron spectroscopy (AES) beyond the ultra-short core-hole relaxation time approximation and, as a test case, we apply it to the paradigmatic example of a one-dimensional Mott insulator represented by a half-filled Hubbard chain. The Auger spectrum is usually calculated by assuming that, after the creation of a core-hole, the system thermalizes almost instantaneously. This leads to a relatively simple analytical expression that uses the ground-state with a core-hole as a reference state and ignores all the transient dynamics related to the screening of the core-hole. In this picture, the response of the system can be associated to the pair spectral function. On the other hand, in our numerical calculations, the core hole is created by a light pulse, allowing one to study the transient dynamics of the system in terms of the pulse duration and in the non-perturbative regime. Time-dependent density matrix renormalization group calculations reveal that the relaxation process involves the creation of a polarization cloud of doublon excitations that have an effect similar to photo-doping. As a consequence, there is a leak of spectral weight to higher energies into what otherwise would be the Mott gap. For longer pulses, these excited states, mostly comprised of doublons, can dominate the spectrum. By changing the duration of the light-pulse, the entire screening process can be resolved in time.

Effective Bi-Layer Model Hamiltonian and Density-Matrix Renormalization Group Study for the High-T c Superconductivity in La3Ni2O7 under High Pressure

High-T c superconductivity with possible T c ≈ 80 K has been reported in the single crystal of La3Ni2O7 under high pressure. Based on the electronic structure given by the density functional theory calculations, we propose an effective bi-layer model Hamiltonian including both 3d z 2 and 3d x 2–y 2 orbital electrons of the nickel cations. The main feature of the model is that the 3d z 2 electrons form inter-layer σ-bonding and anti-bonding bands via the apical oxygen anions between the two layers, while the 3d x 2–y 2 electrons hybridize with the 3d z 2 electrons within each NiO2 plane. The chemical potential difference of these two orbital electrons ensures that the 3d z 2 orbitals are close to half-filling and the 3d x 2–y 2 orbitals are near quarter-filling. The strong on-site Hubbard repulsion of the 3d z 2 orbital electrons gives rise to an effective inter-layer antiferromagnetic spin super-exchange J. Applying pressure can self dope holes on the 3d z 2 orbitals with the same amount of electrons doped on the 3d x 2–y 2 orbitals. By performing numerical density-matrix renormalization group calculations on a minimum setup and focusing on the limit of large J and small doping of 3d z 2 orbitals, we find the superconducting instability on both the 3d z 2 and 3d x 2–y 2 orbitals by calculating the equal-time spin singlet pair–pair correlation function. Our numerical results may provide useful insights in the high-T c superconductivity in single crystal La3Ni2O7 under high pressure.

Electromagnetic signatures of a chiral quantum spin liquid

Quantum spin liquids (QSL) have emerged as a captivating subject within interacting spin systems that exhibit no magnetic ordering even at the lowest temperature accessible experimentally. However, definitive experimental evidence remains elusive. In light of the recent surge in theoretical and experimental interest in the half-filled Hubbard model on a triangular lattice, which offers the potential for stabilizing a chiral QSL, we investigate the electromagnetic signatures of this phase to facilitate experimental detection. Utilizing a combination of parton mean-field theory and unbiased density-matrix renormalization group calculations, we systematically examine the electrical charge and orbital electrical current associated with a spinon excitation in the chiral QSL. Additionally, we calculate the longitudinal and transverse optical conductivities below the Mott gap. Furthermore, employing quantum field theory analysis, we unravel the connection between spinon excitations and emergent as well as physical gauge fields. Our results demonstrate that the chiral QSL phase exhibits a distinct electromagnetic response, even within a Mott insulator regime. This finding holds great potential for enabling the experimental detection of this long-sought-after phase.

Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe_{2}.

The pursuit of exotic phases of matter outside of the extreme conditions of a quantizing magnetic field is a long-standing quest of solid state physics. Recent experiments have observed spontaneous valley polarization and fractional Chern insulators in zero magnetic field in twisted bilayers of MoTe_{2}, at partial filling of the topological valence band (ν=-2/3 and -3/5). We study the topological valence band at half filling, using exact diagonalization and density matrix renormalization group calculations. We discover a composite Fermi liquid (CFL) phase even at zero magnetic field that covers a large portion of the phase diagram near twist angle ∼3.6°. The CFL is a non-Fermi liquid phase with metallic behavior despite the absence of Landau quasiparticles. We discuss experimental implications including the competition between the CFL and a Fermi liquid, which can be tuned with a displacement field. The topological valence band has excellent quantum geometry over a wide range of twist angles and a small bandwidth that is, remarkably, reduced by interactions. These key properties stabilize the exotic zero field quantum Hall phases. Finally, we present an optical signature involving "extinguished" optical responses that detects Chern bands with ideal quantum geometry.

Fermion Disorder Operator at Gross-Neveu and Deconfined Quantum Criticalities.

The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang etal., arXiv:2209.07103]. Here we study, by means of large-scale quantum MonteCarlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.

Observation of a magnetic phase transition in monolayer NiPS3

Monolayer magnet of $\mathit{XY}$ type composes a pivotal part of the two-dimensional magnetism. As an important suspected $\mathit{XY}$-type antiferromagnet, whether $1L \mathrm{NiP}{\mathrm{S}}_{3}$ exhibits magnetic order remains elusive. Herein by helicity-resolved Raman and ultrafast spectroscopy of $\mathrm{NiP}{\mathrm{S}}_{3}$ from bulk to monolayer, we find that $1L \mathrm{NiP}{\mathrm{S}}_{3}$ is magnetically ordered with a Berezinskii-Kosterlitz-Thouless (BKT) transition at ${T}_{\mathrm{BKT}}\ensuremath{\approx}140$ K. We have also performed large-scale density-matrix renormalization-group calculations to verify the ground state zigzag antiferromagnetic order and Monte Carlo simulation to confirm the BKT-transition temperature in $1L \mathrm{NiP}{\mathrm{S}}_{3}$. Our investigations establish $1L \mathrm{NiP}{\mathrm{S}}_{3}$ to be an $\mathit{XY}$ antiferromagnet with a relatively high-temperature BKT transition, providing an important platform for investigating complex couplings and topological excitations.